⚖️ Ratio Province · Percentage change

Reverse percentages

Connect fractions, decimals and multipliers to calculate reverse percentages. In this lesson, focus on percentages compare with 100.

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Understand Reverse percentages

Connect fractions, decimals and multipliers to calculate reverse percentages. In this lesson, focus on percentages compare with 100.

Percentages compare with 100. A reverse percentage starts from the final amount, so divide by the multiplier to recover the original; do not subtract the percentage from the final value. For reverse percentages, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Reverse percentages: Percentages compare with 100. Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount. Keep the reverse percentages representation visible until the final line.

Picture the idea

Fill a 100-cell bar, adjust a multiplier dial and compare original, change and final values. Use the model to explain one change you notice while working on reverse percentages.

Check as you go

Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount. Write that check beside the final reverse percentages answer.

Key vocabulary

percentagemultiplieroriginal valuechangedecimalreversepercentages

Rules and key facts

Given information: Reverse percentages — A coat costs £72 after a 10% discount. Find the original price. Method choice: Turn the percentage into a decimal before using a multiplier. An increase uses a multiplier above 1; a decrease uses a multiplier below 1. Calculation or reasoning: The sale price is 90% of the original. Divide £72 by 0.9 to get £80. Check: 80 × 0.9 = 72. Final answer: 80. Check: An increase uses a multiplier above 1; a decrease uses a multiplier below 1.

  • Identify whether the known amount is original or final.
  • Convert the percentage to a decimal multiplier.
  • Multiply for a forward change or divide for a reverse change.
  • Estimate and label the final answer in context. Record the check explicitly for reverse percentages.

Step-by-step method

  1. Identify whether the known amount is original or final.
  2. Convert the percentage to a decimal multiplier.
  3. Multiply for a forward change or divide for a reverse change.
  4. Estimate and label the final answer in context. Record the check explicitly for reverse percentages.

What you need first

  • Recognise the vocabulary: percentage, multiplier, original value.
  • Be able to explain the purpose of reverse percentages before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Sales and interest
  • Population change

Visual / interactive

See the idea, then move it around

Skip to Practice

Fill a 100-cell bar, adjust a multiplier dial and compare original, change and final values. Use the model to explain one change you notice while working on reverse percentages.

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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Reverse percentages — A coat costs £72 after a 10% discount. Find the original price. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: The sale price is 90% of the original. Divide £72 by 0.9 to get £80. Check: 80 × 0.9 = 72. Final answer: 80. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Identify whether the known amount is original or final.
  2. Convert the percentage to a decimal multiplier.
  3. Multiply for a forward change or divide for a reverse change.
Secure example

Use the normal method

Given information: Reverse percentages — A savings account contains £380 after a 5% increase. Find the amount before the increase. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: After the increase the amount is 105% of the original. Divide £380 by 0.95 to get £400. Check: 400 × 0.95 = 380. Final answer: 400. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Check the reverse percentages result against the original information.

Challenge example

Stretch the idea

Given information: Reverse percentages — A bill including 15% VAT is £322. Find the price before VAT. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: The final bill is 115% of the original. Divide £322 by 1.15 = £280. Check forwards: 280 × 1.15 = 322. Final answer: 280. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Reverse percentages — A final amount is £80 after a 50% discount. Which calculation finds the original amount: 80 ÷ 0.5 or 80 × 0.5? Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: Reverse a percentage change by dividing by its forward multiplier. The correct calculation is 80 ÷ 0.5 = 160. Final answer: 80 ÷ 0.5. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Underline the units and command word. Show the key calculation and write the answer in context.

Common mistakes

  • Subtracting the stated percentage from a final amount in a reverse problem. This is a key trap when answering reverse percentages questions.
  • Using 0.2 instead of 1.2 for a 20% increase.

How to check your answer

Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount. Write that check beside the final reverse percentages answer.

Extension challenge

Create a reverse percentages problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Reverse percentages challenge

Use multiplier battle controls to solve three checked reverse percentages rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Study cards

Core idea

Reverse percentages: Percentages compare with 100. Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount. Keep the reverse percentages representation visible until the final line.

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Key vocabulary

percentage · multiplier · original value · change · decimal · reverse · percentages

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Rules

Identify whether the known amount is original or final. Convert the percentage to a decimal multiplier. Multiply for a forward change or divide for a reverse change. Estimate and label the final answer in context. Record the check explicitly for reverse percentages.

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Formula / fact

Given information: Reverse percentages — A coat costs £72 after a 10% discount. Find the original price. Method choice: Turn the percentage into a decimal before using a multiplier. An increase uses a multiplier above 1; a decrease uses a multiplier below 1. Calculation or reasoning: The sale price is 90% of the original. Divide £72 by 0.9 to get £80. Check: 80 × 0.9 = 72. Final answer: 80. Check: An increase uses a multiplier above 1; a decrease uses a multiplier below 1.

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Foundation example

Given information: Reverse percentages — A coat costs £72 after a 10% discount. Find the original price. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: The sale price is 90% of the original. Divide £72 by 0.9 to get £80. Check: 80 × 0.9 = 72. Final answer: 80. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Reverse percentages — A savings account contains £380 after a 5% increase. Find the amount before the increase. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: After the increase the amount is 105% of the original. Divide £380 by 0.95 to get £400. Check: 400 × 0.95 = 380. Final answer: 400. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Reverse percentages — A bill including 15% VAT is £322. Find the price before VAT. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: The final bill is 115% of the original. Divide £322 by 1.15 = £280. Check forwards: 280 × 1.15 = 322. Final answer: 280. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Reverse percentages — A final amount is £80 after a 50% discount. Which calculation finds the original amount: 80 ÷ 0.5 or 80 × 0.5? Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: Reverse a percentage change by dividing by its forward multiplier. The correct calculation is 80 ÷ 0.5 = 160. Final answer: 80 ÷ 0.5. Check: substitute or compare with the original information to confirm the result fits the question.

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Common mistake

Subtracting the stated percentage from a final amount in a reverse problem. This is a key trap when answering reverse percentages questions.

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Exam tip

For reverse percentages, show the key representation before the final calculation. Use this final check: Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount.

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Real-world use

Sales and interest, Population change

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Checklist

I can explain reverse percentages, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Reverse percentages

Use this whenever a question feels confusing. Nothing here is locked.

Simple explanation

Reverse percentages: Percentages compare with 100. Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount. Keep the reverse percentages representation visible until the final line.

Think of reverse percentages as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Identify whether the known amount is original or final.
  2. Convert the percentage to a decimal multiplier.
  3. Multiply for a forward change or divide for a reverse change.
  4. Estimate and label the final answer in context. Record the check explicitly for reverse percentages.

Hint 1

Start by naming the given information and the exact result required for reverse percentages.

Hint 2

Identify whether the known amount is original or final.

Full worked solution

Given information: Reverse percentages — A coat costs £72 after a 10% discount. Find the original price. Method choice: use the reverse percentages method and show each step with the stated values. Calculation or reasoning: The sale price is 90% of the original. Divide £72 by 0.9 to get £80. Check: 80 × 0.9 = 72. Final answer: 80. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Identify whether the known amount is original or final. → Convert the percentage to a decimal multiplier. → Multiply for a forward change or divide for a reverse change. → Estimate and label the final answer in context. Record the check explicitly for reverse percentages.

Common mistake warning

Subtracting the stated percentage from a final amount in a reverse problem. This is a key trap when answering reverse percentages questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain reverse percentages in my own words.
  • I can use these words accurately: percentage, multiplier, original value.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Subtracting the stated percentage from a final amount in a reverse problem.
  • I can apply this check: Apply the forward multiplier to the answer and confirm that it reproduces the stated final amount.
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